Posted by peeterjoot on January 20, 2014

Here is what will likely be the final update of my class notes from Winter 2013, University of Toronto Condensed Matter Physics course (PHY487H1F), taught by Prof. Stephen Julian.

Official course description: “Introduction to the concepts used in the modern treatment of solids. The student is assumed to be familiar with elementary quantum mechanics. Topics include: bonding in solids, crystal structures, lattice vibrations, free electron model of metals, band structure, thermal properties, magnetism and superconductivity (time permitting)”

This document contains:

• Plain old lecture notes. These mirror what was covered in class, possibly augmented with additional details.

• Personal notes exploring details that were not clear to me from the lectures, or from the texts associated with the lecture material.

• Assigned problems. Like anything else take these as is.

• Some worked problems attempted as course prep, for fun, or for test preparation, or post test reflection.

• Links to Mathematica workbooks associated with this course.

My thanks go to Professor Julian for teaching this course.

NOTE: This v.5 update of these notes is still really big (~18M). Some of my mathematica generated 3D images result in very large pdfs.

Changelog for this update (relative to the first, and second, and third, and the last pre-exam Changelogs).

January 19, 2014 Quadratic Deybe

January 19, 2014 One atom basis phonons in 2D

January 07, 2014 Two body harmonic oscillator in 3D

Figure out a general solution for two interacting harmonic oscillators, then use the result to calculate the matrix required for a 2D two atom diamond lattice with horizontal, vertical and diagonal nearest neighbour coupling.

December 04, 2013 Lecture 24: Superconductivity (cont.)

December 04, 2013 Problem Set 10: Drude conductivity and doped semiconductors.

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Posted by peeterjoot on November 5, 2013

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In phy487 we’ve been using the fact that a periodic function

where

has a Fourier representation

Here is a vector in reciprocal space, say

where

Now let’s express the explicit form for the Fourier coefficient so that we can compute the Fourier representation for some periodic potentials for some numerical experimentation. In particular, let’s think about what it meant to integrate over a unit cell. Suppose we have a parameterization of the points in the unit cell

as sketched in fig. 1.1. Here . We can compute the values of for any vector in the cell by reciprocal projection

Fig 1.1: Unit cell

or

Let’s suppose that is period in the unit cell spanned by with , and integrate over the unit cube for that parameterization to compute

Let’s write

so that

Picking the integral of this integrand as representative, we have when

and when

This is just zero since is an integer, so we have

This gives us

This is our \textAndIndex{Fourier coefficient}. The \textAndIndex{Fourier series} written out in gory but explicit detail is

Also observe the unfortunate detail that we require integrability of the potential in the unit cell for the Fourier integrals to converge. This prohibits the use of the most obvious potential for numerical experimentation, the inverse radial .

Posted in Math and Physics Learning. | Tagged: Fourier coefficient, fourier series, phy487, reciprocal basis, unit cell | Leave a Comment »

Posted by peeterjoot on May 3, 2012

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# Motivation.

I’d used the wrong scaling in a Fourier series over a interval. Here’s a reminder to self what the right way to do this is.

# Guts

Suppose we have a function that is defined in terms of a trigonometric Fourier sum

where the domain of interest is . Stating the problem this way avoids any issue of existence. We know exists, but just want to find what they are given some other representation of the function.

Multiplying and integrating over our domain we have

We want all the terms in the sum to be be zero, requiring equality of the exponentials, or

or

This fixes our Fourier coefficients

Given this, the correct (but unnormalized) Fourier basis for a interval would be the functions , or the sine and cosine equivalents.

# References

Posted in Math and Physics Learning. | Tagged: Fourier coefficient, fourier series | Leave a Comment »